Gate DA 2025 | Question - 9 | General Aptitude

July 15, 2025

If a real variable 𝑥 satisfies $3^{x^{2}} = 27 x 9^{x}$ , then the value of $\frac{2^{x^{2}}}{{(2^{x})}^{2}}$ is:

2⁻¹

2⁰

2³

2¹⁵

Solution:

$27 \times 9^{x} = 3^{3} \cdot (3^{2})^{x} = 3^{3} \cdot 3^{2 x} = 3^{3 + 2 x}$

Now equating both sides:

$3^{x^{2}} = 3^{3 + 2 x}$

Since the bases are the same:

$x^{2} = 3 + 2 x$

$x^{2} - 2 x - 3 = 0$

$(x - 3) (x + 1) = 0$

$x = 3 or x = - 1$

$\frac{2^{x^{2}}}{(2^{x})^{2}}$

Now evaluate this for both values of $x$ :

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